This paper mainly analyzes different analytical solutions of \((2 + 1)\) -dimensional hyperbolic nonlinear Schrodinger equation (HNLS). The problem is employed to model wave propagation, when the dispersion relation’s Hessian is neither strictly positive nor negative. This equation provides a foundational framework for modeling a wide range of physical phenomena, including the propagation of electromagnetic fields, the dynamics of optical soliton transmission, and the evolution of water wave surfaces. In this study, we derive novel and distinct exact solutions of the equation using two effective numerical techniques: the Lie symmetry analysis, and the \(\phi ^6\) -expansion method that have not been reported earlier. Firstly, we apply the \(\phi ^6\) -expansion technique, which presents the exact soliton solutions and also their dynamical structures. The Lie group method reveals nine Lie algebra generators (isomorphism groups), which are employed to investigate the underlying symmetries of the equation. Initially, we determine the associated infinitesimal transformations by applying the one-parameter Lie symmetry approach. Subsequently, we solve the infinitesimal generators to reduce the governing partial differential equation (PDE) into forms with fewer independent variables. These reduced PDEs yield invariant solutions to the governing equation. The resulting exact solutions exhibit diverse dynamic behaviors, including periodic wave solitons, interaction of periodic waves, solitary waves, multisoliton formations, as well as traveling and standing wave profiles. To illustrate these solutions, we present a variety of graphical representations, including two-dimensional (2D), three-dimensional (3D), and contour plots. Furthermore, the exact analytical solutions are computed and verified using symbolic computational tools such as Mathematica. A broad spectrum of novel analytical solutions with distinct dynamical characteristics is analyzed through this approach.
